11.2 Areas of Trapezoids,


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1 11. Areas of Trapezoids, Rhombuses, and Kites Goal p Find areas of other types of quadrilaterals. Your Notes VOCABULARY Height of a trapezoid THEOREM 11.4: AREA OF A TRAPEZOID b 1 The area of a trapezoid is one h half the product of the height b and the sum of the lengths of the bases. A 5 } 1 ( 1 ) Example 1 Find the area of a trapezoid Beavers To prevent beavers from damming a drainage pipe, the trapezoidshaped fence shown is placed at the pipe. Approximate the area enclosed by the fence. 16 ft 1 ft 8 ft 1 ft 8 ft The height of the trapezoid is feet. The lengths of the bases are feet and feet. A 5 } 1 h(b 1 1 b ) Formula for area of a trapezoid 5 1 } ( ) ( 1 ) Substitute. < Approximate. The area enclosed by the fence is about square feet. 98 Lesson 11. Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.
2 11. Areas of Trapezoids, Rhombuses, and Kites Goal p Find areas of other types of quadrilaterals. Your Notes VOCABULARY Height of a trapezoid The height of a trapezoid is the perpendicular distance between its bases. THEOREM 11.4: AREA OF A TRAPEZOID b 1 The area of a trapezoid is one h half the product of the height b and the sum of the lengths of the bases. A 5 } 1 h ( b 1 1 b ) Example 1 Find the area of a trapezoid Beavers To prevent beavers from damming a drainage pipe, the trapezoidshaped fence shown is placed at the pipe. Approximate the area enclosed by the fence. 16 ft 1 ft 8 ft 1 ft 8 ft The height of the trapezoid is 8 Ï } feet. The lengths of the bases are 16 feet and 8 feet. A 5 } 1 h(b 1 1 b ) Formula for area of a trapezoid 5 1 } ( 8 Ï } )( ) Substitute. < 136 Approximate. The area enclosed by the fence is about 136 square feet. 98 Lesson 11. Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.
3 THEOREM 11.5: AREA OF A RHOMBUS Remember that a rhombus is also a parallelogram, so you can also use the formula A 5 b h. The area of a rhombus is one half the product of the lengths of its diagonals. THEOREM 11.6: AREA OF A KITE d d 1 The area of a kite is one half the product of the lengths of its diagonals. d d 1 When you read a diagram, look for information you need to find. The diagram gives the lengths of } QT wand } TR but not the lengths of } QS and } PR. Example Find the area of a rhombus Find the area of the rhombus. P T Step 1 Find the length of each diagonal. The diagonals of a rhombus each other, so QT 5 and PT 5. QS 5 QT cm S 8 cm R 11 cm PR 5 1 TR cm Step Find the area of the rhombus. Let d 1 represent QS and d represent PR. d 1 d Formula for area of a rhombus 5 } 1 ( )( ) Substitute. 5 Simplify. The area of the rhombus is square centimeters. Copyright Holt McDougal. All rights reserved. Lesson 11. Geometry Notetaking Guide 99
4 Remember that a rhombus is also a parallelogram, so you can also use the formula A 5 b h. THEOREM 11.5: AREA OF A RHOMBUS The area of a rhombus is one half the product of the lengths of its diagonals. THEOREM 11.6: AREA OF A KITE The area of a kite is one half the product of the lengths of its diagonals. d d 1 d 1 d d d 1 d 1 d When you read a diagram, look for information you need to find. The diagram gives the lengths of } QT wand } TR but not the lengths of } QS and } PR. Example Find the area of a rhombus Find the area of the rhombus. P T Step 1 Find the length of each diagonal. The diagonals of a rhombus bisect each other, so QT 5 TS and PT 5 TR. QS 5 QT 1 TS cm S 8 cm R 11 cm PR 5 PT 1 TR cm Step Find the area of the rhombus. Let d 1 represent QS and d represent PR. d 1 d Formula for area of a rhombus 5 } 1 ( 16 )( ) Substitute Simplify. The area of the rhombus is 176 square centimeters. Copyright Holt McDougal. All rights reserved. Lesson 11. Geometry Notetaking Guide 99
5 Checkpoint Find the area of the figure ft 4 ft 9 ft. 1 m 18 m Example 3 Solve for unknown measures One diagonal of a kite is two times as long as the other diagonal. The area of the kite is 56.5 square inches. What are the lengths of the diagonals? Draw and label a diagram. Let x be the length of one diagonal. The other diagonal is twice as long, so label it. Use the formula for the area of a kite to find the value of x. x x d 1 d 5 } 1 ( )( ) Substitute. 5 Simplify. Formula for area of a kite 5 x Find the positive square root of each side. The lengths of the diagonals are inches and ( ) 5 inches. 300 Lesson 11. Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.
6 Checkpoint Find the area of the figure ft 4 ft 3 ft 9 ft. 1 m 43 m 18 m Example 3 Solve for unknown measures One diagonal of a kite is two times as long as the other diagonal. The area of the kite is 56.5 square inches. What are the lengths of the diagonals? Draw and label a diagram. Let x be the length of one diagonal. The other diagonal is twice as long, so label it x. Use the formula for the area of a kite to find the value of x. x x d 1 d Formula for area of a kite } 1 ( x )( x ) Substitute x Simplify x Find the positive square root of each side. The lengths of the diagonals are 7.5 inches and ( 7.5 ) 5 15 inches. 300 Lesson 11. Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.
7 Example 4 Find an area in the coordinate plane Yard You have a diagram of your backyard. Each square represents a 3 meter by 3 meter square. Find the area of your backyard. y B C Step 1 Find the lengths of the bases and the height of trapezoid ABCD. b 1 5 BC 5 5 m b 5 AD 5 5 m h 5 BE 5 5 m Step Find the area of ABCD. 3 A 3 E D x A 1 m by 15 m rectangle with an area of 315 m would enclose your yard. So the answer of 5 m is reasonable. h(b 1 1 b ) 5 1 } ( )( 1 ) 5 The area of your backyard is square meters. Checkpoint Complete the following exercises. 3. One diagonal of a kite is three times as long as the other diagonal. The area of the kite is 73.5 square yards. What are the lengths of the diagonals? Homework 4. Find the area of a rhombus with vertices M(, 4), N(5, 6), P(8, 4), and Q(5, ). Copyright Holt McDougal. All rights reserved. Lesson 11. Geometry Notetaking Guide 301
8 Example 4 Find an area in the coordinate plane Yard You have a diagram of your backyard. Each square represents a 3 meter by 3 meter square. Find the area of your backyard. y B C Step 1 Find the lengths of the bases and the height of trapezoid ABCD. b 1 5 BC m b 5 AD m h 5 BE m Step Find the area of ABCD. 3 A 3 E D x A 1 m by 15 m rectangle with an area of 315 m would enclose your yard. So the answer of 5 m is reasonable. A 5 } 1 h(b 1 1 b ) 5 } 1 ( 15 )( ) 5 5 The area of your backyard is 5 square meters. Checkpoint Complete the following exercises. 3. One diagonal of a kite is three times as long as the other diagonal. The area of the kite is 73.5 square yards. What are the lengths of the diagonals? 7 yards and 1 yards Homework 4. Find the area of a rhombus with vertices M(, 4), N(5, 6), P(8, 4), and Q(5, ). 1 square units Copyright Holt McDougal. All rights reserved. Lesson 11. Geometry Notetaking Guide 301
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